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Some Finite Series

  • Natural numbers: \(n\), \(k\)
    Number of terms of a series: \(n\)
    1. Sum of the first \(n\) natural numbers
      \(1 + 2 + 3 + \ldots + n =\) \({\large\frac{{n\left( {n + 1} \right)}}{2}\normalsize}\)
    2. Sum of the first \(n\) even natural numbers
      \(2 + 4 + 6 + \ldots + 2n =\) \({n\left( {n + 1} \right)} \)
    3. Sum of the first \(n\) odd natural numbers
      \(1 + 3 + 5 + \ldots\) \(+ \left( {2n – 1} \right) =\) \( {n^2}\)
    4. Sum of \(n\) natural numbers starting from \(k\)
      \(k + \left( {k + 1} \right) + \left( {k + 2} \right) + \ldots\) \(+ \left( {k + n – 1} \right) =\) \({\large\frac{{n\left( {2k + n – 1} \right)}}{2}\normalsize}\)
    5. Sum of the squares of the first \(n\) natural numbers
      \({1^2} + {2^2} + {3^2} + \ldots + {n^2} =\) \({\large\frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}\normalsize}\)
    6. Sum of the cubes of the first \(n\) natural numbers
      \({1^3} + {2^3} + {3^3} + \ldots + {n^3} =\) \({\left[ {{\large\frac{{n\left( {n + 1} \right)}}{2}}\normalsize} \right]^2}\)
    7. Sum of the squares of the first \(n\) odd natural numbers
      \({1^2} + {3^2} + {5^2} + \ldots\) \( + {\left( {2n – 1} \right)^2} =\) \({\large\frac{{n\left( {4{n^2} – 1} \right)}}{3}\normalsize}\)
    8. Sum of the cubes of the first \(n\) odd natural numbers
      \({1^3} + {3^3} + {5^3} + \ldots\) \( + {\left( {2n – 1} \right)^3} =\) \({n^2}\left( {2{n^2} – 1} \right)\)
    9. \({\large\frac{1}{{1 \cdot 2}}\normalsize} + {\large\frac{1}{{2 \cdot 3}}\normalsize} + {\large\frac{1}{{3 \cdot 4}}\normalsize} + \ldots\) \(+ {\large\frac{1}{{n \left( {n + 1} \right)}}\normalsize} =\) \({\large\frac{n}{{n + 1}}\normalsize} =\) \(1 – {\large\frac{1}{{n + 1}}\normalsize}\)
    10. \({\large\frac{1}{{1 \cdot 2 \cdot 3}}\normalsize} + {\large\frac{1}{{2 \cdot 3 \cdot 4}}\normalsize} + {\large\frac{1}{{3 \cdot 4 \cdot 5}}\normalsize} + \ldots\) \( + {\large\frac{1}{{n\left( {n + 1} \right)\left( {n + 2} \right)}}\normalsize} =\) \({\large\frac{1}{2}\normalsize}\left[ {{\large\frac{1}{2}\normalsize} – {\large\frac{1}{{\left( {n + 1} \right)\left( {n + 2} \right)}}}\normalsize} \right]\)